A synthetic proof (without coordinates) of Pascal’s theorem
is possible with the aid of cross ratios or the related notion
of harmonic sets (of four collinear points).

Pascal’s proof is lost; presumably he had only the real affine plane
in mind. A proof restricted to that case, based on Menelaus’s theorem,
can be seen at
http://www.cut-the-knot.org/Curriculum/Geometry/Pascal.shtml#wordscut-the-knot.org.